· 1. introduction to high energy astrophysics 3 1introduction to high energy astrophysics 1.1aims...

1

Astronomy 345: High Energy Astrophysics I

Session 2017-1811 Lectures, starting September 20, 2017

Lecturer: Dr E. KontarKelvin Building, room 615, extension x2499

Email: Eduard (at) astro.gla.ac.uk

Lecture notes and example problems - https://moodle2.gla.ac.uk

last update: November 4, 2017

High Energy Astrophysics I

https://moodle2.gla.ac.uk

CONTENTS 2

Contents

1 Introduction to High Energy Astrophysics 3

2 Observing X-rays 24

3 X-ray emission mechanisms 33

4 Black-body spectrum 45

5 Reaction cross section 55

6 Thomson scattering 64

7 Bremsstrahlung 74

8 Thermal and Multi-thermal Bremsstrahlung 89

9 Photon spectrum interpretation 103

10 Inverse Compton Scattering 114

11 Synchrotron Radiation 131


1. INTRODUCTION TO HIGH ENERGY ASTROPHYSICS 3

1 Introduction to High Energy Astrophysics

1.1 Aims To introduce students to the physical processes responsible for X-

Ray production, as a basis for the applications discussed in X-RayAstrophysics II

To introduce students to the concept of a reaction cross-section, andto explain how to calculate X-Ray emission rates and spectra fromspecified source conditions

Further details are also available in Course Hand-book


http://www.gla.ac.uk/coursecatalogue/course/?code=ASTRO4009http://www.gla.ac.uk/coursecatalogue/course/?code=ASTRO4009


1.2 Recommended literature and useful resources:These lecture notes are following the material from:

Malcolm S. Longair, High energy astrophysics,volume 1 & 2, High energy astrophysics, 1992, e.g.Amazon


http://www.amazon.com/High-Energy-Astrophysics-Particles-Detection/dp/0521387736http://www.amazon.com/High-Energy-Astrophysics-Particles-Detection/dp/0521387736


1.3 A Brief History of X-rays

Wilhelm Roentgen

Oct 1895 Wilhelm Roentgen begins tostudy Cathode Rays (discovered decadesearlier)

8 Nov 1895 Roentgen notices glowing flu-orescent screen some distance away fromcathode ray tube - realises he has discov-ered a new phenomenon: X-rays

22 Dec 1895 Roentgen photographs hiswifes hand - The first X-Ray Picture



1.3.1 First X-ray Picture

Print of Wilhelm Rontgens first X-ray, of his wifes hand, taken on 22December 1895 and presented toLudwig Zehnder of the Physik Insti-tut, University of Freiburg, on 1 Jan-uary 1896



1.3.2 X-ray timeline: 28 Dec 1895 Discovery announced at Wurzburg Physico-Medical

Society

4 Jan 1896 Discovery announced at Berlin Physical Society

Jan 1896 Discovery published in newspapers around the world

2 Mar 1896 Henri Becquerel discovers natural radioactivity of Ura-nium



1.4 From the past till now 1901 First ever Nobel Prize in Physics awarded to Roentgen

1903 Third Nobel Prize in Physics awarded to Becquerel, Pierre andMarie Curie

...

1999 Launch of CHANDRA and XMM X-ray satellites

2002 Launch of RHESSI X-ray and gamma-ray satellite

2008 Launch of Fermi X-ray and gamma-ray satellite



1.5 X-ray Region of Electromagnetic SpectrumWavelength :

0.01 nm.. 10 nm

0.1 .. 100

where 1 nm= 109 m, 1 = 1010 m.Corresponding photon energy:

E = h= hc

= 6.6310343108

1.61019 [m] [eV]=12.4 []

[keV]

where 1 eV= 1.6021019 J.

1 keV= 103 eV, 1 MeV= 106 eV, 1 GeV= 109 eV, 1 TeV= 1012 eV



1.6 Classification/terminologyEnergy range Wavelength range

Soft X-rays 0.11 keV 10010 Classical X-rays 110 keV 101 Hard X-rays 10100 keV 10.1 Gamma-rays (-rays) & 0.1 MeV . 0.1

Note that the classification/terminology is somewhat different in vari-ous areas of Astrophysics.

Energy Temperature:

E ' kBT = 1 keV' 107 K

where kB = 1.381023 J/K Boltzmann constant.Hence to produce X-rays we need very high temperatures!



1.7 X-ray and atmosphereOnly a few windows in the E-M spectrum exist for ground-based ob-

servations: optical and radio (see Figure 1).

Figure 1: X-ray penetration via atmosphere



Space Technology opened up rest of E-M Spectrum.Classical X-rays:

These are readily absorbed (photoelectric absorption) by gases, liq-uids and solids

Unable to penetrate the Earths atmosphere

Observations must be made at altitudes above 100 km using rocketsor satellites

Hard X-rays:

More penetrating than classical X-rays. Observations can be madefrom e.g. balloon platforms at 30 km altitude

Soft X-rays

Much weaker flux than classical or hard X-rays. Strongly attenuatedby interstellar gas in the galaxy



1.8 Useful wavelength range:Useful range for obtaining astronomical information:

1021 m.. 104 m

For . 1021 m (Energy & 1015 eV), -rays readily destroyed bycollisions with CMBR photons, producing electron-positron pairs

For & 104 m, radio waves are absorbed by solar wind plasma (cut-off frequency near Earth 20 kHz) . Earth ionosphere producescut-off near 10 MHz



1.9 X-Ray Astronomy from SpaceRocket experiments started with captured German V2 rockets. (flight

lasted for only a few minutes)

1948 Solar X-rays detected (from the solar corona)

1962 Bright X-ray source discovered in Scorpius; star denoted ScoX-1

1963 Isotropic X-ray background discovered; Extragalactic Sources;X-ray source detected in Crab Nebula

1966 X-ray galaxies identified



1.10 Satellite MissionsSince 1970 till present day large number of satellites launched to ob-

serve various astrophysical objects in X-rays e.g. ROSAT, Yohkoh, SoHo,XMM, Chandra, RHESSI,...

Huge numbers of sources detected (e.g. 60000 by ROSAT; > 10times more by XMM, Chandra)

Many X-ray binaries identified (e.g. Cygnus X-1, black hole candi-date)

Detailed observations of solar flares, pulsars, supernova, galaxies,quasars, galaxy clusters, moon, comets...



Figure 2: ROSAT X-ray bright sources see ROSAT webpage


http://www.dlr.de/dlr/en/desktopdefault.aspx/tabid-10424/


Figure 3: All sky map of gamma-ray counts above 1 GeV from NASAFermi data


http://fermi.gsfc.nasa.gov/ssc/http://fermi.gsfc.nasa.gov/ssc/


Figure 4: Abell galaxy cluster by Chandra X-ray observatoryHigh Energy Astrophysics I

https://www.nasa.gov/mission_pages/chandra/main/index.html


Figure 5: X-ray jet blasting out of the nucleus of M87 Chandra


http://chandra.harvard.edu/photo/2001/0134/


Figure 6: X-ray image of Crab nebula Chandra, Harvard


http://chandra.harvard.edu/photo/1999/0052/


Figure 7: GOES soft X-ray flux from the Sun. see data from NOAA SpaceWeather Prediction Center


http://legacy-www.swpc.noaa.gov/rt_plots/xray_5m.htmlhttp://legacy-www.swpc.noaa.gov/rt_plots/xray_5m.html


Figure 8: SDO andRHESSI observa-tions of a solar flare: 106 K coronaplasma (yellow),10 keV X-ray (red),> 30 keV X-rays(blue) from Astron-omy & Astrophysics


http://sdo.gsfc.nasa.gov/http://hesperia.gsfc.nasa.gov/rhessi3/http://www.aanda.org/articles/aa/full_html/2011/09/aa17605-11/aa17605-11.htmlhttp://www.aanda.org/articles/aa/full_html/2011/09/aa17605-11/aa17605-11.html


Figure 9: X-rays are produced by fluorescence when solar X-rays bom-bard Moon Chandra, Harvard


http://chandra.harvard.edu/photo/2003/moon/index.html

2. OBSERVING X-RAYS 24

2 Observing X-rays

2.1 Grazing Incidence Optics

Figure 10: Grazing Incidence Optics

For low energy photons E .


7. BREMSSTRAHLUNG 79

7.3 Bremsstrahlung Cross-Section

Bremsstrahlung cross-section can be written

dQBd

(,E)= Q0mc2

Eln

(1+p1/E1p1/E

)(7.3)

This is known as the Bethe-Heitler formula, where

Q0 = 83r2e = 1.541031 [m2]

recalling that = 1/137, we find that

Q0 = QT137The Bethe-Heitler formula is valid in the regime where the initial vi and

final v f velocities of the electron satisfy:

c vi, v f cHigh Energy Astrophysics I


Figure 35: Log-term ln(

1+p1/E1p1/E

)from Eq. 7.3.

(For lower vi velocities we require quantum mechanical corrections; forhigher velocities we require relativistic corrections)

Note that the term

ln

(1+p1/E1p1/E

)



is rather complicated (see Fig 35).This factor should be included in precise calculations (e.g. RHESSI),

but it varies fairly slowly with and we will replace it by a constant.

For simplicity we can take,

ln

(1+p1/E1p1/E

)= 1

so that bremsstrahlung cross-section can be written

dQBd

(,E)= Q0mc2

E[m2keV1] (7.4)

This is known as the Kramers approximation.



7.4 Optically thin spectrumSubstituting Equation (7.4) into the expression for differential emissiv-

ity (Eq. 7.1), one finds a simplified expression:

d jd

= npQ0mc2

F(E)E

dE [ photons m3s1keV1]

For an extended, optically thin source of volume V , in which we havenp = np(r ), F(E)= F(E,r ), i.e. proton number density and electron en-ergy distribution are, in general, functions of position, the bremsstrahlungdifferential emissivity is (using Kramers approximation)

dJd

= Q0mc2

V

np(r )

F(E,r )E

dEdV [ photons s1keV1]

(7.5)



and the differential luminosity

dLd

= dJd

is given by

dLd

=Q0mc2

Vnp(

r )

F(E,r )E

dEdV [ J s1keV1] (7.6)

We will apply these formulae later, and in the example sheets.



7.5 Bremsstrahlung luminosity spectrumSuppose there exists some non-zero energy, Emin, such that the ki-

netic energy of all electrons satisfies E > Emin.It then follows that F(E,r ) = 0, for all E < Emin, and the energy inte-

gral in Equation 7.6

F(E,r )E

dE =

Emin

F(E,r )E

dE

Hence, for all photon energies < Emin, the differential emissivitydJd

= Q0mc2

V

np(r )

Emin

F(E,r )E

dEdV [ photons s1keV1]

and the luminosity

dLd

=Q0mc2

Vnp(

r )

Emin

F(E,r )E

dEdV [ J s1keV1]



Note that dLd is independent of the photon energy .Hence at low photon energies, < Emin, the differential bremsstrahlung

spectrum, dLd , is flat.

Figure 36: For a thermal distribution of electrons, we have F(E) 0 as E 0,so we can make the approximation F(E,r ) = 0, for all E < Emin and and hence theluminosity spectrum of thermal bremsstrahlung becomes flat at low photon energies.



7.6 Non-thermal bremsstrahlungConsider a non-Maxwellian distribution of electron velocities - e.g. a

power law differential electron flux spectrum:

F(E)= F0E

where F(E)dE is the fraction (or number) of particles with energy be-tween E and E+dE, and F0 and .

Note that the power law is usually defined from some low energy cutoff, Emin, because a power law extending to E 0 would formally giveF(E) for > 0.

Consider electrons with kinetic energy E > Emin. Total flux for > 1

Ftot =

Emin

F(E)dE =

Emin

F0EdE = F01E1

Emin

= F01E

1min

i.e. we find the constant

F0 = Ftot(1)E1minHigh Energy Astrophysics I


where Ftot is the flux above Emin.Differential emissivity with power-law flux spectrum of electrons is,

then:dJd

= Q0mc2

V

np(r )

F0E

EdEdV

dJd

= Q0mc2

V

np(r )

F0E1dEdV

which is valid for E > Emin.For a uniform plasma with np(

r ) = const = np and F0, constantthroughout the volume,

dJd

= Q0mc2

npV F0

(E

)



dJd

= Q0mc2

npV F0(+1) (7.7)

hencedJd

(+1); dLd

= dJd

Thus, the non-thermal (collisional) bremsstrahlung photon spectrumis also a power law, with the same exponent as the power law differentialelectron flux spectrum.

Observationally, by measuring the slope of the observed photon spec-trum, we can deduce the slope of the electron flux spectrum.


8. THERMAL AND MULTI-THERMAL BREMSSTRAHLUNG 89

8 Thermal and Multi-thermal BremsstrahlungConsider an isothermal, homogeneous plasma of fully ionised hydro-

gen. We have ne = np independent of position

F(E)= FM(E)= v(E)dnedEwhere dnedE is the Maxwellian distribution. Hence the flux spectrum

FM(E)=

2Em

2npp

E1/2

(kT)3/2exp

( E

kT

)Note: before we defined F(E) = F(E,r ), this is equivalent to writingF(E)= F(E,T) provided we can define T = T(r ).

The differential emissivity is, then

dJd

= Q0mc2

V

np(r )

F(E,T)E

dEdV =High Energy Astrophysics I


= 2n2pVQ0mc

2

2m

EE

1(kT)3/2

exp( E

kT

)dE

Put z = E/kT and integrate to obtain the emissivity spectrum

dJd

= 2

2m

n2pVQ0mc2

(kT)1/2exp

(

kT

)(8.1)

and luminosity spectrum

dLd

= 2

2m

n2pVQ0mc2

(kT)1/2exp

(

kT

)(8.2)

Thus, the energy spectrum for thermal bremsstrahlung from a homo-geneous plasma decays exponentially at high photon energies.



8.1 Spectral shapeSpectrum which falls expo-nentially at high energies -thermal, see e.g. Figure 21.Observationally, measuringthe shape of dLd allows us todetermine a temperature, T,for the plasma. We can dothis, e.g., for the X-ray emis-sion from galaxy clusters andthe Sun. (The isothermal as-sumption may break down,however).



8.2 Thermal bremsstrahlung from Coma cluster

Figure 37: XMM-Newton mosaic image of the central region of Coma (5overlapping pointings) in the [0.3-2] keV energy band and isothermal withwith kT = 8.25 keV from Arnaud et al, 2001


http://adsabs.harvard.edu/abs/2001A%26A...365L..67A


8.3 Thermal (and non-thermal) bremsstrahlung from solar flares

Figure 38: Left: RHESSI image of a solar flare: red - thermal, blue - non-thermal, yellow background is 1 MK plasma from SDO/AIA; Right: X-rayspectrum of a limb flare. From Kontar et al, 2010


http://adsabs.harvard.edu/abs/2010ApJ...717..250K


8.4 Multi-Thermal BremsstrahlungRecall Equation 8.2 for thermal bremsstrahlung:

dLd

= 2

2m

n2pVQ0mc2

(kT)1/2exp

(

kT

)Measuring dL/d also permits us to determine n2pV . This quantity isknown as the emission measure.

Note that the mass of gas in the plasma is given by (ignoring the massof electrons): Mgas = npV mp.

So the emission measure does not directly tell us Mgas; to determinethis we need to estimate V separately.

For example, for the X-ray emission from a galaxy cluster, we canassume the cluster is spherical and use its angular size and redshift toestimate its volume.



8.5 Inhomogeneous plasmaSuppose the plasma is not isothermal. Consider the differential emis-

sivity per unit volume at position, r (from Equation 8.1)

d jd

= 2

2m

n2p(r )Q0mc2

(kT(r ))1/2 exp(

kT(r )

)[ photons m3s1keV1]

Integrated emissivity for the whole volume is then

dJd

=

V

d jd

dV

i.e.dJd

= 2

2m

Q0mc2

k1/2

V

n2p(r )

T(r )1/2 exp(

kT(r )

)dV



8.6 Source emission measure functionOne useful approach to simplifying this expression is to replace the

integral over volume with an integral over temperature. We do this byintroducing the source emission measure function 2. This is a measure ofhow much of the plasma is at temperature T.

We define T(T)dT =

V

n2p(r )dV (8.3)

where r =r (T).From which it can be shown that (e.g. using integration by parts)

dJd

= 2

2m

Q0mc2

k1/2

0

(T)1

T1/2exp

(

kT

)dT (8.4)

2sometimes this value is called differential emission measure



8.7 Spherical volumeWe can see more easily how this substitution works for the specific

example of a spherically symmetric temperature distribution, i.e.

T = T(r)

We make the change of variables from (r,,) to (T,,) and we write

dV = r2 sin()dddr = dSdr

where dS is area element.Then make a substitution:

dr = drdT

dT = dTdTdr



It follows thenV

n2p(r )

T(r )1/2 exp(

kT(r )

)dSdr =

T

S

n2p(r )

T(r )1/2 dTdr exp(

kT(r )

)dS dT =

=T

(T)1

T(r )1/2 exp(

kT(r )

)dT

where

(T)=

ST

n2p(r )dT

dr

dSST denotes spherical surface of constant temperature T, at radius r.

Things simplify further if the plasma density is also spherically sym-metric i.e. np(

r )= np(r).3

3Problem sheet works through an example in detail



8.8 Isothermal surfaces

Figure 39: Isothermal surfaces

More generally T = T(r,,) but wecan still change variables in the integralby identifying isothermal surfaces - i.e.surfaces of constant temperature (whichwill not in general be spherical).

The source emission measure is nowdefined in terms of the temperature gra-dient, T

(T)=

ST

n2p(r )

|T| dS

but we will not consider any non-spherical cases here.



8.9 Examples of inhomogeneous plasmaThe differential emissivity from non-uniformly heated plasma can be

presented using (T), see Equation 8.4:

1

dLd

= dJd

= 2

2m

Q0mc2

k1/2

0

(T)1

T1/2exp

(

kT

)dT

A wide range of behaviour for (T) is possible:

(T) smoothly varying in an extended gas cloud (e.g. solar corona,galaxy cluster Fig 40)

(T) has a sharp change or discontinuous across a shock front (e.g.transition region in the solar atmosphere, supernova remnant, Fig.41)



Figure 40: Temperature structure of the galaxy cluster Abell 3921. FromBelsole et al, 2005


http://cdsads.u-strasbg.fr/cgi-bin/bib_query?2005A&A...430..385B


Figure 41: CassiopeiaA Supernova Remnant:In this false-color image,NuSTAR data, which showhigh-energy X-rays fromradioactive material, arecolored blue. Lower-energyX-rays from non-radioactivematerial, imaged previouslywith NASAs Chandra X-rayObservatory, are shown inred, yellow and green fromNuSTAR webpage


http://www.nustar.caltech.edu/image/nustar140219ahttp://www.nustar.caltech.edu/image/nustar140219a

9. PHOTON SPECTRUM INTERPRETATION 103

9 Photon spectrum interpretation

9.1 Mimicking Non-thermal ProcessesWe saw earlier that, if the non-thermal differential electron flux distribu-

tion is a power law, then the differential photon luminosity is also a powerlaw, with the same exponent.

If, instead, we have a thermal plasma, but not an isothermal plasma,then we can also obtain a power law photon spectrum - provided thesource emission measure function has a power law dependence on tem-perature, i.e. 4

if (T) T then dLd

with 6= , but there is ambiguity between thermal and non-thermal pro-cesses.

4See problem sheet



9.2 Interpreting Energy SpectraThe fact that measurements of dJ/d and dL/d are not perfect, but

are subject to experimental errors, raises some important numerical is-sues.

The problem is to determine, from dJ/d:

(T) for a thermal source

F(E) for a non-thermal source

Consider a non-thermal source, homogeneous plasma:

dJd

= Q0mc2

npV

F(E)E

dE



The integral

F(E)E

dE

is a function of photon energy, . We define

G()=

F(E)E

dE

It follows thatdJd

= Q0mc2

npVG()

Then

G()= 1Q0mc2npV

dJd



Let us differentiate G()5

dGd

= dd

F(E)E

dE = F(E)E

E=

then we can find

F(E)=(dGd

)=E

=E dGd

=E

and substituting expression for G(), we have

F(E)= EQ0mc2npV

dddJd

=E

= EQ0mc2npV

[dJd

+ dd

(dJd

)]=E(9.1)

5If f(y) is a function that is continuous on our interval, then

ddx

b(x)a(x)

f (y)d y= f (b(x)) dbdx

f (a(x)) dadx

where a(x) and b(x) are functions of x.



Since the solution (9.1) for F(E) involves the derivative of the photonenergy spectrum, dJd , this means that small changes in the measureddata for dJd can lead to large changes in the inferred F(E).

6

To solve the problem we need to apply regularization. Extraction ofelectron spectrum from photon spectrum is a challenge, see e.g. Brownet al, 2006.

6This an example of an ill-posed inverse problem.


http://adsabs.harvard.edu/abs/2006ApJ...643..523Bhttp://adsabs.harvard.edu/abs/2006ApJ...643..523B


9.3 Example: Hard X-ray spectrum of a solar flare

Figure 42: Albedo-corrected RHESSI spectrum (crosses with error bars) at the hard X-ray peak of the solar flare on 2002-06-02 from Holman et al, 2011. The solid line showsthe combined isothermal (dotted line) plus double power-law (dashed line) spectral fit.The spectral fit before albedo correction is over-laid (gray, solid line).


http://link.springer.com/article/10.1007/s11214-010-9680-9


9.4 Example: Derivative of a noisy data

Figure 43: A power-law spectrum J(E) 1/E (left panel) and the abso-lute value of the derivative |dJ/dE| 1/E2 (right panel); without and with3% Gaussian noise added.



9.5 Derivative as an inverse problemLet us assume that we have a smooth function J(E) over the interval

E01 E E02. We have a finite sample Ji of measured values of thisfunction, obtained over some grid E01 = E0 < E1 < ...< E i < ...< En = E02with mesh size E. The noisy data set has an error

|Ji J(E i)| J (9.2)

where J is an uncertainty of measurement.We want to find the best smooth estimate of the derivative J (E) using

the given data set E [E01,E02]. The two point finite difference estimateis readily available with the following boundJi+1 JiE J (E i)

O(E+J/E), (9.3)where the first and second terms in the right hand side represent consis-tency and propagation errors respectively.



9.6 Thermal X-ray Emission LinesHot astrophysical plasmas in e.g. the Sun contain traces of heavier

elements which are partially (often highly!) ionised.

Figure 44: Simulated X-ray spectra of solar flare plasma thermal emissionfor a range of plasma temperatures from 10 to 50 MK, from Skinner et al,2013


http://adsabs.harvard.edu/abs/2013arXiv1311.6955Shttp://adsabs.harvard.edu/abs/2013arXiv1311.6955S


Figure 45: The Sun at171 chiefly emitted by FeIX and X at million degreesK. Elements produce emis-sion lines, superimposed ona thermal background (Fig23). Such emission lines canbe observed using narrow fil-ters, sensitive to only a nar-row range of X-ray energies(temperatures).



Figure 46: SDO/AIA observations of multi-temperature corona.


http://sdo.gsfc.nasa.gov/

10. INVERSE COMPTON SCATTERING 114

10 Inverse Compton Scattering

The scattering of photons in media due to their interaction with elec-trons.

Thompson scattering (which we already discussed in Lecture 6) is thespecial case of Compton scattering in the limit of a low energy incomingphoton - i.e. in the rest frame of the electron, the photon has incomingfrequency , such that

h mc2

In this low energy limit, the frequency , of the outgoing photon satisfies

=

and the reaction cross-section is equal to the Thompson cross-section(see Section 6). This is purely classical result.



10.1 Compton ScatteringMore generally (Compton, 1923, Klein & Nishina, 1929), we need to

take into account :

1) Modification of cross-sectionThompson cross-section replaced by Klein-Nishina formula

QK N =QT[

12 hmc2

+ 256

(h

mc2

)2 ...

]

2) Electron recoils, and absorbs some of the photons energy , so that

>

e.g. energy loss by a photon.


http://adsabs.harvard.edu/abs/1923PhRv...21..483Chttp://adsabs.harvard.edu/abs/1929ZPhy...52..853K


10.2 Inverse Compton ScatteringA low energy photon collides with a relativistic electron and gains en-

ergy at the expense of the electron (e.g. radio photons might be boostedto X-ray energies).

So for Inverse Compton scattering we want

E E,

In most astrophysical situations it is still OK to assume that, in the restframe of the electron, the energy of the photon before collision is muchless than the rest mass energy of the electron i.e.:

h mc2

This means that we do not need to use the Klein-Nishina formula, but canassume

QIC QT = 83r2e



10.3 Kinematics (head-on collision)To study the kinematics of inverse Compton scattering, we consider

first the case of a head-on collision.

Figure 47: Photon of energy , scattered through 180o by a relativisticelectron of initial energy E1.



Recall from special relativity, energy and momentum together makeup the 4-vector, 4-momentum and Lorentz factor is = 1p

1v2/c2For the photon:

energy: = hmomentum: p =

c= h

cFor electron:

energy: E = mc2momentum: p = mv

Let us recall the useful relation, for the electron

E2 = p2c2+m2c4hence

p2 = E2

c2m2c2 = 2m2c2m2c2 = (21)m2c2



= p = mc21

By the conservation of energy, we have

E1+1 = E2+2E1E2 =E = 21 (10.1)

and conservation of momentum:

p1 1c = p2+2

c(10.2)

We want to express 2 (final photon energy) in terms of 1 and E1, i.e.we want to eliminate E2.

We introduce the dimensionless energy variable for the photon

= mc2

analogous to

= Emc2



Equations (10.1, 10.2) give

12 =21 = (10.3)2111 =

221+2 (10.4)

Thus2 = 1

211221= 1+2 = 21+

Substituting for 2:211

(1)21= 21+

= (1)21=[

211 (21+)]2

simplifying, one obtains

= [1+21

211

]= 21

[2111

]High Energy Astrophysics I


=21

[2111

][1+21

211

] (10.5)We can further simplify this formula by noting that, for astrophysical

examples of inverse Compton scattering, we have:

Low energy incoming photons: 1 1High energy incoming electrons: 1 1

so we have from Equation (10.5)

=21

[2111

][1+21

211

] = 211[

11/211/1]

1

[1+21/1

11/21

]Let us retain first order terms in 1/1, and using that (1+ x)n ' 1+nx, we



can derive

= 21[11/2211/1

][1+21/11+1/221

] ' 2121[211+1/2

] = 4121[411+1

] (10.6)Equation 10.6 can be further simplified. However, since we assumed 1 1, and 1 1, the value 11 is undefined, i.e. 11 1 or 11 1

Let us first consider the case 11 1 in 10.6:

' 4121[

411+1] ' 1 (10.7)

Equation 10.7 is approximation saying that photon gains all the energy ofthe incoming electron.

However, even larger relative boost can be achieved if 11 1.



Let us now consider the case 11 1 in Equation (10.6), so we canwrite:

' 4121 (10.8)also since = 21 ' 2

2 ' 4211 (10.9)

Note that a head-on collision gives the maximum energy transfer tothe outgoing photon.

Averaging over all scattering directions gives approximately:

2 ' 43211 (10.10)



10.4 Example ILet us consider head-on collision between an optical photon 1 = 1 eV

and a cosmic ray electron.Note that 1 = 1/mc2 = 103/511' 2106.



10.5 Example IIFor a CMBR photon 1 = 103 eV and a cosmic ray electron. Note that

1 = 1/mc2 = 106/511' 2109.



10.6 Validity of approximationsTwo examples show that in two cases:

Head-on collision between a CMBR photon and a cosmic ray elec-tron

Head-on collision between an optical photon and a cosmic ray elec-tron

In all cases we considered, 11 1, as we assumed in deriving theexpression for 2.



10.7 Inverse Compton LuminosityConsider a homogeneous volume, V , filled with electrons all of energy

E = mc2 and number density ne, and photons all of energy = h andnumber density n

Total emissivity from this volume is given by (see Equation 5.2):

J = NTFQWe regard the photons as the target particles, hit by a beam of highly

relativistic electrons. Thus:

NT = nV and F = nev ' necso that

J = nV necQIC (10.11)From previous section, average energy of a scattered photon,

2 ' 432h



Thus, average source luminosity from Equation (10.11)

L = J2 = nV necQIC 432h

We can re-write this as follows:

L IC = 43c2 Ne=neV

QIC U=nh

, [W] (10.12)

where U is the radiation energy density.Hence, the average IC power emitted from Ne electrons of energymc2 (

dEdt

)IC

= L IC = 43QICcNe2U [W] (10.13)



10.8 Lifetime of an electron and IC lossesPower emitted by a single electron

L IC = 43QICc2U

Timescale for an electron to lose its energy

IC EdEdt

= mc2

329 r

2ec2U

= 9mc32r2e

1U

Let us consider e.g., CMBR photons T ' 2.7 K. From black body radiationenergy density (see Equation 4.6 and description in Section 4)

U = aT4 ' 41014, [J m3]Timescale for an electron to lose its energy in CMBR background

IC 9mc32r2e1U

' 21012

[ years]



For, e.g., 1 GeV electron

= Emc2

' 2103

we have IC ' 109 years.


11. SYNCHROTRON RADIATION 131

11 Synchrotron Radiation

11.1 IC spectrumIn a real situation the electrons and photons would have a distribu-

tion of initial energies, on which the Inverse Compton cross-section, QICwould depend - i.e. for an IC photon of outgoing energy = h we have

dQICd

= dQICd

(h0,E)

where h0 is the energy of incoming photon, E is the energy of incomingelectron. Further

dJICd

= c

dnedE

dnd0

dQICd

d0dEdV

We can define the electron energy before collision via = E/mc2, so thatwe can also write

dJICd

= c

dned

dnd0

dQICd

d0ddV



anddL IC

d= c

dned

dnd0

dQICd

d0ddV

We can simplify this integral by make the assumption that all emittedphotons gain the average amount of energy i.e.

dQICd

= 0 unless = 432h0

we can write this as

dQICd

= 83r2e

(432h0

)=QT

(432h0

)(11.1)

where (x) is Dirac delta function.7

7 The Dirac delta function has the following properties:

(x x0)= 0 if x 6= x0

(x x0)dx = 1 and

f (x)(x x0)dx = f (x0)



Changing variables from to x = 4/32h0 at fixed 0, we have:dL IC

d= c

dned

dnd0

QT(x)d0 38h0dxdV

Integrating over the electron energy x

dL ICd

= cQT (

dned

38h0

)x=

dnd0

d0dV

Substituting back = x = 432h0 and simplifying, we find luminosity spec-trum of Inverse Compton emission for arbitrary electron and photon spec-tra

dL ICd

= cQT (

2dned

)=

3

4h0

dnd0

d0dV , [ W keV1] (11.2)

Now we need to assume the spectrum of incoming electrons.High Energy Astrophysics I


11.2 Power-law spectrum of electronsSuppose we have a power-law electron spectrum independent of po-

sition i.e.dned

= K (11.3)

Then (

2dned

)=

3

4h0

= K2+1 = K

2

(3

4h0

)12

so luminosity differential in energy

dL ICd

= c83r2e

K2

(3

4h

)12

1

2

12

0dnd0

d0dV , [ W keV1] (11.4)

i.e. if we neglect the constants



dL ICd

, where = 12

(11.5)

So, again, we find that a power-law distribution of electron energiesgives rise to a power law photon spectrum, but with a different powerlaw index.



11.3 Summary of power-law spectraX-ray mechanism Electron distribution Photon index

Non-thermal Bremsstrahlung F(E) E dLd

Thermal inhomogeneous plasma T dLd

Inverse Compton scattering dned dLd (1)

2

Note spectrum for synchrotron radiation in the next section.



11.4 Cyclotron radiation

Figure 48: Electron gy-rating in

B field

Consider a non-relativistic electron, with v c,following a circular orbit or radius, r, around afield line of a uniform magnetic field,

B .

Lorentz force gives circular motion (Figure 48)

evB = mv2

r= L = vr =

eBm

where L is Larmor angular frequency. Electronemits cyclotron radiation at the Larmor frequencyL

L = L2 =eB

2m(11.6)

Any constant velocity component parallel to the magnetic field does notlead to radiation (no change in acceleration, recall Equation 6.2).



11.5 Cyclotron line features

Figure 49: Hercules X-1 X-ray spectrum from Trumper etal, 1978

Cyclotron line has energy:

= hL = heB2m ' 107B, [ keV Tesla1]

Cyclotron lines are observed in X-ray bina-ries due to resonant scattering of the line ofsight X-ray photons against electrons embed-ded in magnetic fields.

For e.g. Her X-1 (X-ray binary) has suchfeature near 37 keV, (discovered by Trumperet al, 1978 see also Furst et al, 2013) so candiagnose magnetic field:

B ' 3.7108, [ Tesla]

Note that the strong magnetic fields are expected near to a neutron star.High Energy Astrophysics I

http://adsabs.harvard.edu/abs/1978ApJ...219L.105Thttp://adsabs.harvard.edu/abs/1978ApJ...219L.105Thttp://adsabs.harvard.edu/abs/1978ApJ...219L.105Thttp://adsabs.harvard.edu/abs/1978ApJ...219L.105Thttp://adsabs.harvard.edu/abs/2013ApJ...779...69F


11.6 Synchrotron RadiationConsider now a highly relativistic electron v c, and 1.Radiation is known as synchrotron and is strongly Doppler shifted and

forward beamed due to relativistic aberration (Figure 50).

Figure 50: Cartoon showing relativistic beaming of synchrotron radiation



11.7 Synchrotron frequency and single electron spectrum

Figure 51: Synchrotron spectrum of a singleelectron

Typical frequency of syn-chrotron radiation is

s = 322L = 32

2(

eB2m

)(11.7)

Synchrotron radiation is emit-ted over a wide range of fre-quencies (Figure 51).

Peak occurs at 0.3s,but average frequency value '

s. At low frequencies, s, the spectrum grows 1/3 (Figure 51).



11.8 Synchrotron luminosityWe can estimate the power radiated by a single electron using Equa-

tion 6.2:

P = e2v2

60c3

where the acceleration v can be determined by transforming to the elec-trons rest frame, in which electric field is:

E = v B

From Newtons 2nd law

dv

dt= e

mE = e

mv B

We can estimate the power radiated by a single electron is(dE

dt

)S= e

2v2

60c3= e

2

60c3

(evBsin()

m

)2(11.8)



where dE/dt is the power in electron rests frame. But the power is anLorentz invariant, e.g.

dE

dt= dE

dtHence the observed power radiated per electron is also given by the for-mula 11.8.

If v is randomly oriented in 3-D, then sin2() = 23.Also, the magnetic energy density is defined to be

UB = B2

20

where 0 is permeability constant. Recalling Thomson cross-section (Equa-tion 6.4)

QT = 83(

e2

40mc2

)2= 8

3r2e



the power emitted by an electron (Equation 11.8) can be re-written:(dEdt

)S= e

2

60c323

(evB

m

)2= 4

3QT c2UB (11.9)

where we took v = c and 00 = 1/c2.

If we compare this formula (Equation 11.9) with our result for the In-verse Compton luminosity (Equation 10.13), one finds:(dE

dt

)IC(dE

dt

)S

= UUB

(11.10)

The ratio of Inverse Compton and Synchrotron luminosity from asource is given by the ratio of its radiation to magnetic energy density.



11.9 Synchrotron radiation from electron distribution

For a homogeneous volume, V , of uniform magnetic field,B , con-

taining Ne electrons - all of energy E = mc2 then (as for the InverseCompton case) the total luminosity for Ne electrons is given by

LS = 43QT cNe2UB [ W] (11.11)

As we considered in the Inverse Compton case, in a real situation theelectrons would have a distribution of energies, and their number densitywould be a function of position. We need to take this into account whenwe calculate the synchrotron spectrum - i.e. the luminosity as a functionof photon energy. Thus, synchrotron luminosity spectrum is

dLSd

=

43

c2dned

UBdQSd

ddV (11.12)



11.10 Synchrotron luminosity spectrumTo simplify matters, we make a similar approximation as in the Inverse

Compton case (see Lecture 10): we assume that synchrotron emissiononly occurs at the mean synchroton frequency

s = 322L

and all synchrotron photons have energy:

= hs = 322hL

This means that we assume the synchrotron differential cross-sectiontakes the form:

dQSd

= 83r2e

(322hL

)(11.13)

Hence, we can write the differential synchrotron spectrum as

dLSd

= 43

c

2dned

UB83r2e

(322hL

)ddV



and integrating over , one finds 8

dLSd

= 43

cQT (

3hL

dned

)=

2

3hL

UBdV (11.14)

8Here one can use another property of Dirac delta function

( f (x))= 1| f (a)|(xa),

where a is the root of f (x) i.e. f (a)= 0.High Energy Astrophysics I


11.11 Radiation from a power-law electron spectrumSuppose we have a power-law electron spectrum independent of po-

sition i.e.dned

= K, > 0

Then assuming uniform distribution of electrons and uniformB :

dLSd

= 43

cQT(

3hLK

)=

2

3hL

UBdV (11.15)

ordLSd

12which is the same result as for Inverse Compton Scattering.



11.12 Typical energies of synchrotron electrons and photonsWe saw previously that, for extremely high magnetic fields (e.g. near

to a pulsar), we can obtain X-ray cyclotron emission/absorption:

= hL = heB2m ' 107B [keV/Tesla]

since s = 3/22L we could in principle achieve X-ray synchrotron ener-gies for more modest magnetic fields, provided is large enough.

For e.g. the Solar corona, fields of about 10-100 Gauss 9 have beenmeasured. Thus to observe X-ray synchrotron emission, with e.g. 10 keV,from the corona would in this case require

322107102 = 10

hence2 ' 1010 = v = 0.99999999995c

9Note that 1 Gauss =104 Tesla



so v should be very close to c!Suppose instead we take a more modest v = 0.5c

2 = (10.25)1 = 43

= 1.52hL ' 2107B [keV/Tesla]Thus for B = 102 Tesla=100 Gauss, = 2106 eV or

L = 21061.61019

6.631034 ' 480 [ MHz]

This is in the radio part of the E-M spectrum. Indeed, gyro-synchrotronemission peaking near 10 GHz is often observed during solar flares(e.g. Figure 52), indicating mildly relativistic particles 0.1-1 MeV.



11.13 Solar energetic particles and gyro-synchrotron emission

Figure 52: Simulations (centre) of the spectrum (right) and the images(left) of radio emission provides strong evidence that the emission mech-anism is (gyro-) synchrotron radiation - due to the acceleration of chargedparticles in the Sun s magnetic field. From Nita et al, 2015


http://adsabs.harvard.edu/abs/2015ApJ...799..236N

Introduction to High Energy AstrophysicsObserving X-raysX-ray emission mechanismsBlack-body spectrumReaction cross sectionThomson scatteringBremsstrahlungThermal and Multi-thermal BremsstrahlungPhoton spectrum interpretationInverse Compton ScatteringSynchrotron Radiation

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· 1. introduction to high energy astrophysics 3 1introduction to high energy astrophysics 1.1aims...

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